Everest, Graham; Röttger, Christian; Ward, Thomas B. The continuing story of zeta. (English) Zbl 1228.11128 Math. Intell. 31, No. 3, 13-17 (2009). Summary: We show how the binomial theorem can be used to continue the Riemann zeta-function to the left hand half-plane. This method yields the explicit values of the function at non-positive integers in terms of the Bernoulli numbers. Cited in 1 ReviewCited in 5 Documents MSC: 11M06 \(\zeta (s)\) and \(L(s, \chi)\) 11B68 Bernoulli and Euler numbers and polynomials Keywords:binomial theorem; continuation of the Riemann zeta-function; explicit values of the Riemann zeta-function at non-positive integers; Bernoulli numbers PDFBibTeX XMLCite \textit{G. Everest} et al., Math. Intell. 31, No. 3, 13--17 (2009; Zbl 1228.11128) Full Text: DOI arXiv References: [1] T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976. · Zbl 0335.10001 [2] R. Ayoub, Euler and the Zeta Function, Amer. Math. Monthly 81 (1974), 1067–1086. · Zbl 0293.10001 [3] R. Dwilewicz and J. Mináč, The Hurwitz zeta function as a convergent series, Rocky Mountain J. Math. 36 (2006), 1191–1219. · Zbl 1193.11080 [4] G. R. Everest and T. Ward, An Introduction to Number Theory, Springer-Verlag, Graduate Texts in Mathematics Vol. 232, New York, 2005. · Zbl 1089.11001 [5] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th edition, The Clarendon Press, Oxford University Press, New York, 1979. · Zbl 0423.10001 [6] E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 3rd edition, Chelsea Publishing Company, 1974. · JFM 40.0232.08 [7] J. Mináč, A remark on the values of the Riemann zeta function, Exposition. Math. 12 (1994), 459–462. · Zbl 0812.11051 [8] M. Ram Murty and M. Reece, A simple derivation of {\(\zeta\)}(1 k) k /k, Funct. Approx. Comment. Math. 28 (2000), 141–154. · Zbl 1034.11048 [9] S. J. Patterson, An Introduction to the Riemann Zeta-Function, Cambridge Studies in Advanced Mathematics 14, Cambridge University Press, Cambridge, 1988. · Zbl 0641.10029 [10] A. van der Poorten, A proof that Euler missed ... Apéry’s proof of the irrationality of {\(\zeta\)}(3). An informal report. Math. Intelligencer 1, no. (1978/79), 195–203. [11] K. Prachar, Primzahlverteilung, Grundlehren 91, Springer, Berlin, 1957. [12] V. Ramaswami, Notes on Riemann’s zeta function, J. London Math. Soc. 9 (1934), 165–169. · Zbl 0009.34801 [13] G. F. B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie, November (1859). [14] J. Sondow, Analytic continuation of Riemann’s zeta function and values at negative integers via Euler’s transformation of series, Proc. Amer. Math. Soc. 120 (1994), 421–424. · Zbl 0796.11033 [15] B. Sury, Bernoulli numbers and the Riemann zeta function, Resonance 8 (2003), 54–62. [16] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, edited with a preface by D. R. Heath-Brown, second edition, The Clarendon Press, Oxford University Press, New York, 1986. · Zbl 0601.10026 [17] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. · Zbl 0951.30002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.